Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals

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1 Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals Samuel B. Trickey Sept Quantum Theory Project Dept. of Physics and Dept. of Chemistry

2 So Is B3LYP the Answer? Given that it has only 3 parameters and performs remarkably well, is B3LYP about as good as as we are going to do? Quite some time ago, Ruiz, Salahub, and Vela [J. Phys. Chem. 100, (1996)] responded in the negative: The B3LYP results lie between those of the GGA and MP2. [Our results] for the so-called half-and-half potential are in very good agreement with those obtained through second-order Møller-Plesset calculations and with available experimental data. However, the more widely used three-parameter, B3LYP, functional does not perform well; the hybrid methods are not a panacea. They studied C 2 H 4 X 2 (X=F, Cl, Br, or I) complexes in three orientations See their table, next slide. LDA = f ( ρ ) xc ρ ρ r ( )

3 So Is B3LYP the Answer? LDA = f ( ρ ) xc ρ ρ r ( ) Ruiz, Salahub, and Vela [J. Phys. Chem. 100, (1996)]

4 So Is B3LYP the Answer? Errors in transition-metal -ligand bond dissociation energies (BDE, in kj mol 1 ) of methyl and carbene complexes of first-row transition metal cations BDE by CI methods MCPF, CCSD(T) always too small, sometimes significantly (carbenes!) Curves for B3LYP and PCI-80(MP2) quite similar in shape [PCI-80: correlation energy (from MP2 or CCSD) scaled by an empirical factor] For M-CH 3+, PCI-80 uniformly very good; B3LYP in general too large, second best For M-CH 2+, PCI-80 often somewhat too large; B3LYP overall best Credit: N. Rösch

5 Ways Forward? - KLI There is a relatively fast way to do ExX approximately with fairly high acccuracy, the Krieger-Lee-Iafrate (KLI) approximation. Suppose we had a good E c functional to go with this approximate ExX. Is that a way forward? Here is a sobering table. F under method is KLI ExX. NONE of the ExX + E c combinations does better than GGA! Finding an E c to go with ExX is not easy.

6 Ways Forward? Local Hybrid Functionals Hybrid functionals mix exact (single-determinant) exchange, E xks, with approximate E x and E c contributions in fixed proportions: E = λ E + (1 λ) E + E xchybrid xks x, approx c, approx Why not do the mixing locally, i.e., pointwise? Here is the Jaramillo et al. version of the idea [J. Chem. Phys. 118, 1068 (2003) ( ) λ ( ) ( ) 1 λ ( ) ( ) ( ),, ( ) { } ExcLocHybrid = d r n r r u r xks + r u r x approx + u r c approx λ ( r) ( r ) n ( r W ) τ ( r) ( r) 8n( r) W τ : = 1 ; : = τ Recall (Lect. V-3, 8, II-27): τ W / τ is a so-called iso-orbital indicator. Results next slide. 2

7 Ways Forward? Local Hybrid Functionals (cont d.) Dissociation of H 2 + for BLYP, B3LYP, and lh-blyp XC G++(3df,3pd) basis. [J. Chem. Phys. 118, 1068 (2003)

8 Ways Forward? Range-separated Hybrids Rather than mix exact (single-determinant) exchange, E xks, with approximate E x on a local basis, what about separating the Coulomb potential into a short- and long-ranged part and treating them separately to get a hybrid? [A. Savin et al., Internat. J. Quantum Chem. 56, 327 (1995); Chem. Phys. Lett. 275, 151 (1997)] Motivation: approximate E xc functionals are local or semi-local, so use them at short range and use exact E xks at long range. Here is the range-separated Coulomb interaction: 1 1 γ ( ri j ) 1 γ ( ri j ) g( rij ) = = + : = glr ( rij ) + r r r r r Vˆ i j i j 1 ˆ 1 = g ( r ) ; V = gsr( rij ) 2 2 ee, LR LR ij ee, SR i j 2 γ ( ri j ) = erf ( µ ri j ) ; erf ( x) : = dx e π i j i j i j x 0 x 2 g SR ( r ) ij

9 Ways Forward? Range-separated Hybrids (cont d.) F And here is the range-separated universal functional and its minimum: [ n] : = min ψt ˆ + Vˆ + Vˆ RS ee, LR ee, SR ψ n { } = min ψt ˆ + Vˆ ψ + min ψt ˆ + Vˆ ψ min ψt ˆ + Vˆ ψ ee, LR ee ee, LR ψ n ψ n ψ n [ ] [ ] min ψt ˆ Vˆ Vˆ ψ [ ] E = F n + E n = E n 0 RS 0 ext 0 ee, LR ee, SR ext 0 ψ ψ { } E [ n ] = min ψt ˆ + Vˆ ψ + min min ψt ˆ + Vˆ ψ min ψt ˆ + Vˆ ψ + ee, LR ee ee, LR ext 0 ψ n ψ n ψ n Usual E v cxt f unctional LR version of E v cxt [ ] ψtˆ Vˆ, F n = min + ψ + F [ n] RS ee LR SR ψ n

10 Ways Forward? Range-separated Hybrids (cont d.) One version of a range-separated hybrid LDA, makes the following approximations [Gerber et al., J. Chem. Phys. 127, (2007)] [ ] F Φ ; n = Φ T ˆ + Vˆ Φ + F [ n] RS ee, LR SR 1 Φ ( 1,, Ne ) = det ϕ 1 ϕn N! e 1/3 SR, µ uxlda ( r) = A( r) π erf(1/ 2 A( )) π 2 8 r A ( r) = ( ( ) ( ) ) ( ) e ( ) ( ) ( ) } + 2 A 4 A exp 1/ (4 A ) 3 A + 4 A r r r r r 1/3 ( π 2 ) n 1/3 ( r) 2 3 µ The SR xlda comes from short-ranged HEG work of Toulouse, Savin, and Fladd [Internat. J. Quantum Chem. 100, 1047 (2004)]. In the calculations shown on the next slide, Gerber et al. used VWN for E c.

11 Ways Forward? Range-separated Hybrids (cont d.) HSE= Heyd, Scuseria, Ernzerhof hybrid Remark though this is good work based on a very clever idea, it isn t obvious that the results are a major improvement on ordinary hybrids. [Gerber et al., J. Chem. Phys. 127, (2007)]

12 Ways Forward? Range-separated Hybrids (cont d.) PBE0= PBE-based hybrid [Gerber et al., J. Chem. Phys. 127, (2007)]

13 Ways Forward? Range-separated Hybrids (cont d.) From the Abstract of Gerber et al. [J. Chem. Phys. 127, (2007)] The RSHX functional, which has the main feature of providing a correct asymptotic behavior of the exchange potential, has a tendency to improve the description of structural parameters with respect to local and generalized gradient approximations. The band gaps are too strongly opened by the presence of the long-range Hartree-Fock exchange in all but wide-gap systems. In the difficult case of transition metal oxides, the gap is overestimated, while magnetic moments and lattice constants are slightly underestimated. Note: Hartree-Fock exchange is again a misnomer, even though they work with an orbital-dependent (non-local) potential.

14 Adiabatic Connection Now comes one of the more powerful concepts for understanding functionals and constructing them. The objective is a smooth transformation from the noninteracting KS system to the fully interacting physical system. (This is distinct from the smooth mixing that is in range-separated and local hybrid functionals.) The first ingredient is the Hellmann-Feynman theorem. Suppose the Hamiltonian depends smoothly on a parameter λ. Then, the ground state is H ˆ Ψ = E Ψ ; Ψ Ψ = 1 Differentiate w/r to the parameter λ λ,0 λ,0 λ,0 λ,0 λ,0 E,0 Ψ ˆ,0 ˆ H ˆ Ψ λ λ λ,0 = H λ λ Ψ λ,0 + Ψ λ,0 Ψ λ,0 + Ψ λ,0 H λ λ λ λ λ Hˆ Hˆ λ λ = Eλ,0 Ψλ,0 Ψ λ,0 + Ψλ,0 Ψ λ,0 = Ψλ,0 Ψ λ,0 λ λ λ The second ingredient is the Pauli coupling constant trick. Given the Hamiltonian H ˆ = H ˆ + λh ˆ λ 0 1 it follows from the Hellmann-Feynman theorem that 1 E E = d λ ψ Hˆ ψ λ= 1 λ= 0 0 λ 1 λ

15 Adiabatic Connection (cont d.) Go back to Levy-Lieb constrained search but for a one-parameter Hamiltonian as in the Pauli trick. Then the universal functional is F [ n] = min ψ Tˆ + λv ψ λ ψ n λ ee λ λ = 0: F [ n] = min Φ Tˆ Φ Φ Tˆ Φ T [ n] 0 λ Φ n min,n min,n s λ = 1: F [ n] = min ψ Tˆ + V ψ ψ Tˆ + V ψ 1 ψ n λ= 1 ee λ= 1 min, n ee min, n λ= 1 Therefore F [ n] F [ n] = E [ n] + E [ n] 1 0 xc ee By invoking a Lagrange multiplier potential v λ ( r ) which keeps the density UNchanged across the whole range 0 λ 1, and using the Hellmann-Feynman theorem and Pauli trick, one can prove the adiabatic connection 1 E [ n] = dλ ψ [ n] Vˆ ψ [ n] E [ n] xc 0 λ, min ee λ, min ee Notice that (a) v-representability is back in the picture, both non-interacting and interacting and (b) we don t have to know the potential v λ ( r ).

16 Ways Forward? Adiabatic Connection Functionals The adiabatic connection can be rewritten simply to a suggestive form Some facts are known about the W functional: [ ] W 0, n = E [ n] xks W W = E [ n], 2 nd -order Goerling-Levy C GL 0 c,2 λ 0 W < 0 λ λ [ ] 1 1 E ˆ xc[ n] = dλ ψ 0 λ, min[ n] Vee ψ λ, min[ n] Eee[ n] : = dλw λ, n 0 A scheme for calibrating approximate interpolation λ=0 λ= 1 based on approximate E xc also is known. [Cohen, Mori-Sánchez, and Yang, J. Chem. Phys. 127, (2007)] W [ λ, n] = Exc[ n1/ λ ( r) ] λ 2 Ex, approx [ n] + 2 λec, approx[ n1/ λ ( r) ] + λ Ec, approx[ n1/ λ ( r) ] λ n 1/ λ 3 ( r) = λ n( r / λ )

17 Ways Forward? Adiabatic Connection Functionals (cont d.) The Mori-Sánchez, Cohen, and Yang [J. Chem. Phys. 124, (2006)] model adiabatic-connection path is λb n, { ϕ} W [ λ, n] = a n, { ϕ } λc n, { ϕ} b b ln ( 1+ c) Exc n, { ϕ} = a + c c 2 The choice of BLYP as the interpolating functional, for example, gives ( c) c ln 1+ Exc n, { ϕ} = ExKS + W 0 2 c BLYP ExKS λw 0 + Wλ p c = BLYP λ E W p ( xks λ ) p A selected summary of results [Cohen, Mori-Sánchez, and Yang, J. Chem. Phys. 127, (2007)] follows. The original paper has many very large tables.

18 Ways Forward? Adiabatic Connection Functionals (cont d.) Remark we are back to the no free lunch theorem. These are intricate functionals that require extensive work to generate and program, yet give comparatively modest improvement over simple hybrids. [Cohen, Mori-Sánchez, and Yang, J. Chem. Phys. 127, (2007)]

19 Ways Forward? Back to the GGA?!? Armiento and Mattsson [Phys. Rev. B 72, (2005)] produced a GGA that behaves properly for two model systems, the HEG and the jellium surface and, in effect, interpolates between them by measuring the local inhomogeneity. The resulting functional has a structure that looks a lot like PBE but is different in content. Results are impressive [J. Chem. Phys. 128, (2008)] Consideration of better ways to constrain (hence parameterize) GGAs is an area of active research.

20 A Digression: One-shot Functionals Irrespective of the choice of E xc approximation, it often is desirable to get an estimate of the DFT energy for multiple nuclear configurations {R} of some system, without, doing the full scf calculation. The Harris approximate, non-iterative functional is the best known of several one-shot ways to do this. [J. Harris, Phys. Rev. B 31, 1770 (1985)] Suppose that one has a reasonably good approximate density, n A. Then 2 { 1 ( 2 ) + vext ( )] + vee ( )] + vxc ( )]} h [ n ( r)] φ = [ n r [ n r [ n r φ = ε φ (1) (1) (1) (1) KS A k A A A k k k 1 n ( r) n ( r ) E n d d E n ( 1) A A Harris[ A( r) ] = ε k r r + xc[ A( r) ] drvxc[ na( r) ] k 2 r r The rough physical reasoning is that the KS equation was derived from variational stability. Therefore, a non-self-consistent solution of it should lead to an error reduction which is embodied in the resulting eigenvalues ε i (1) The other terms handle over-counting.

21 Ways Forward Some Commentary It is probably fair to say that a majority of the DFT functional development community believes explicitly orbital-dependent functionals, beyond the level of MGGAs, are a necessity. However, the evolution of increasingly sophisticated hybrids seems to be reaching a state of diminishing returns. A minority points to M06-L and AM05 as examples to argue that orbitaldependence beyond the level of MGGAs is, in fact, not necessary. The agenda, therefore, of this minority, is to find MGGA and similar functionals with hybridlevel performance. The use of full ExX is still hampered by the lack of really good approximate E c functionals to accompany E xks

Advanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University

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